1. Field of the Invention
The present invention relates to a correlation detection method and a connection-structure estimation method for identifying connections and estimating a connection structure among neurons and their synaptic strengths which are required for construction of a neural network model which attracts much research attention in recent years. The methods are based upon time-dependent correlation among data pertaining to a plurality of neurons.
2. Description of the Prior Art
In recent years, research activities on neural-network models are done extensively, resulting in a large number of proposed neural-network models. These research activities are aimed at the development of a neuro-computer for implementing functions which are difficult to realize with the von Neumann computer.
At the present time, however, the performance of such a neural-network model is still not satisfactory in comparison with the information-processing capability of animals. Therefore, it is strongly believed that neural network models based on the study of information processing mechanism of animals' nervous systems should have better performance.
In order to construct a model of a neuronal network of nervous systems, it is the most important to clarify the structure of the neuronal network of nervous system and estimate the synaptic strengths among a large number of neurons based on results of their activities measurement. Therefore, a variety of techniques for detecting connectivities among neurons from time-course data of the neurons have been tried so far.
FIG. 1 shows a flowchart of a typical conventional technique for detecting connectivities among neurons described in a paper that appears on pages 403 to 414 of the Biological Cybernetics magazine, Issue No. 57 published in 1987.
As shown in the figure, the flowchart comprises three steps ST1, ST2 and ST3. At the step ST1, data is input. At the step ST2, a crosscoincidence histogram is calculated. At the step ST3, a scaled crosscoincidence histogram is calculated.
Next, operations performed in accordance with the flowchart are explained. First of all, at the ST1, action potential train data of two neurons X and Y are input. Then, at the step ST2 a histogram CCH.sub.xy (t) of time difference of action potentials between neurons X and Y is calculated from the data input at the step ST1. This histogram CCH.sub.xy (t) is referred to hereafter as a crosscoincidence histogram.
Subsequently, at the step ST3, the crosscoincidence histogram CCH.sub.xy (t) calculated at the step ST2 is normalized into a scaled crosscoincidence histogram SCCH.sub.xy (t) in order to eliminate effects of the measurement time and the number of action potentials. The scaled crosscoincidence histogram SCCH.sub.xy (t) is given by Eq. 1 as follows: EQU SCCH.sub.xy (t)=T.times.CCH.sub.xy (t)/(N.sub.x .times.N.sub.y .times..DELTA.)-1 (1)
where T is the measurement time, N.sub.x is the number of action potentials of the neuron x, N.sub.y is the number of action potentials of the neuron Y and DELTA is the width of the unit time used in the histogram.
The following three problems are encountered in the analysis of action potential train data of actual neurons by means of the conventional technique for detecting connectivities among neurons described above. The first problem is that the detection sensitivity is bad due to a poor correlation S/N ratio. The second problem is that since the scaled crosscoincidence histogram does not depend on the synaptic strength in a simple way, it is difficult to estimate the synaptic strength from the scaled crosscoincidence histogram. The third problem is that since the scaled crosscoincidence histogram depends on the frequency of action potential of presynaptic neuron, it is impossible to distinguish variations of the scaled crosscoincidence histogram caused by changes in synaptic strength from variations caused by changes in frequency of action potential.
In particular, it is important to put the third problem into consideration when analyzing a learning mechanism which is expected as an essential function of a neural network model. The learning of nervous systems is given rise to by the change in synaptic strength. Accordingly, in order to analyze a learning mechanism, it is necessary to estimate the change in synaptic strength among neurons by a learning process.
It is extremely difficult to analyze the learning mechanism by the correlation detection method which depends not only on synaptic strength but also on the frequency of action potentials of a presynaptic neuron, because such method cannot estimate the change in synaptic strength.
A concrete example is given by explaining results of a study by simulation of a neuron model. The simulation results show how a scaled crosscoincidence histogram depends upon the synaptic strength and the frequency of action potential of a presynaptic neuron.
In this example, neuronal activities of neurons X and Y shown in FIG. 7 are simulated by an equation expressed by Eq. 2 below. This equation which was proposed by Hodgkin and Huxley in 1952 has been generally used because it describes neuronal activities and action potential phenomena most accurately. ##EQU1## where I is a membrane current, C is a membrane capacitance, V is a membrane potential, g.sub.Na, g.sub.k, g.sub.L and g.sub.Syn are maximum conductances of a sodium current, a potassium current, other ion currents and a synaptic current, E.sub.Na, E.sub.K, E.sub.L and E.sub.syn are equilibrium potentials of the sodium current, the potassium current, the other ion currents and the synaptic current, m, h and n are parameters of sodium activation, sodium inactivation and potassium activation and f(t) is a function expressing variations in synaptic current with the lapse of time.
The first and the other terms on the right side of Eq. 2 represent a capacitance current and ion currents, respectively. Especially the fifth term represents the synaptic current.
In addition, the parameters m, h and n representing activities of ion currents can each also be expressed by a differential equation and depends on a membrane potential. For example, a differential equation for the parameter m representing activation of sodium currents is given by Eq. 3 as follows: EQU dm/dt=(1-m).times..alpha.-m .beta. EQU .alpha.=0.1.times.(-35-V)/(exp((-35-V)/10)-1) EQU .beta.=4.times.exp ((-60-V)/18) (3)
Likewise, differential equations for the parameters h and n are also expressed similarly to Eq. 3. By solving four differential equations, i.e. Eq. 2 and the equations for the parameters m, h and n, variations in membrane potential of the neurons and the generation of action potentials can be computed. FIG. 2 shows a scaled crosscoincidence histogram computed from action potential trains of the neurons X and Y obtained by simulation. As shown in the figure, at a time delay of 11 ms, a correlation peak is detected. Then the simulation and the computation of the scaled crosscoincidence histogram were performed by varying the synaptic strength and the frequency of action potentials.
FIG. 3 shows the relation between synaptic strength and the peak value of scaled crosscoincidence histograms which are calculated from action potential trains obtained by each simulation. The horizontal axis of the figure represents relative synaptic strengths in a neuron model. The curves represent relations between the synaptic strength and the peak value for different action potential frequency.
It is obvious from the results of simulation of a neuron model shown in FIG. 3 that even for a given synaptic strength, the peak value is different from histogram to histogram depending upon action potential frequency of the neuron X.
In order to propose a new correlation detection method, we pay attention to the information theory which has not been taken notice in the neurological field. In the information theory, a quantity called mutual information is particularly focused on. This quantity represents an information flowing between two processes statistically. It should be noted that the mutual information has relation with the synaptic strength between neurons.
In general, it is believed that connectivity between two neurons X and Y can be detected by the mutual information. Eq. 4 given below was derived in order to calculate mutual information from time-course data of action potentials of two neurons X and Y: EQU I(X:Y.sub.t)=.SIGMA.p(Y.sub.j,t .vertline.X.sub.i)p(X.sub.i) log (p(Y.sub.j,t .vertline.X.sub.i)/p(Y.sub.j,t)) (4)
where I() is the mutual information p (Y.sub.j,t .vertline.X.sub.i) is a conditional probability that the state of the neuron Y is Y.sub.j,t at a time difference of t when the state of the neuron X is X.sub.i and p (X.sub.i) and p (Y.sub.j,t) are probabilities of occurrence that the state of the neuron X is X.sub.i and the state of the neuron Y is Y.sub.j respectively. The summation on the right side of Eq. 4 is carried out for the states of the neurons X and Y.
FIG. 4 shows a computation result of the mutual information for the same train of action potentials as FIG. 2. In comparison with FIG. 2, the peak relative to the background is higher, so mutual information detects the correlation more sensitively than the scaled crosscoincidence histogram.
The conventional technique using mutual information for detecting a connectivity between neurons is implemented by following the procedure described above. The mutual information is a function of frequency of action potentials of presynaptic neuron, p (X.sub.i), as shown by Eq. 4. Therefore, the conventional technique using mutual information has a problem that it is difficult to distinguish variations caused by changes in synaptic strength from variations caused by changes in frequency of action potentials as well as the conventional method using scaled crosscoincidence histogram. In addition, the conventional technique using mutual information has another problem that it is impossible to distinguish an inhibitory synaptic connectivity from an excitatory synaptic connectivity because the mutual information is always nonnegative.
Next, conventional techniques related to the method of estimation of a connection structure among neurons is described.
In order to analyze the dynamic behavior of a nervous system from multi-point recording data, first of all, it is necessary to detect connections among neurons and to estimate the connection structure among the neurons. That is why connectivities among neurons have to be computed and the connection structure among the neurons must be estimated. By merely calculating the connectivity between two neurons, however, it is impossible to distinguish a direct synaptic connectivity between the two neurons from an indirect connectivity dependent on activities of another neuron. In order to solve this problem, a variety of techniques for distinguishing a direct synaptic connectivity between the two neurons from an indirect connectivity have been tried.
FIG. 5 shows a flowchart of the conventional technique for estimating a connection structure among neurons described on pages 987 to 999 of the Biophysical Journal, Issue No. 57, published in 1990. As shown in the figure, the flowchart comprises a data-input step ST4, a crosscoincidence-histogram-calculation step ST5, a probability-calculation step ST6 and a step ST7 at which a connectivity between two neurons independent of a third one is calculated.
The following is a description of operations of the conventional technique. First of all, at the step ST4, time-course data of action potentials of three neurons X, Y and Z is input. Subsequently, at the step ST5, crosscoincidence histograms CCH.sub.XY (t), CCH.sub.xz (t), CCH.sub.YZ (t) among the three neurons X, Y and Z are computed. Here, t represents a time difference of the neuron Y with the neurons X and Z taken as a reference. Next, at the step ST6, these crosscoincidence histograms are normalized and joint probabilities p(X,Y.sub.t,Z), p(X,Z), p(Y.sub.t,Z), are calculated.
Then, a connectivity h(t) between the neurons X and Y is estimated by using Eq. 5 given below. EQU h(t)=log ((p(X,Y.sub.t,Z)p(Z))/(p(X,Z)p(Y.sub.t,Z)))-logr* (5)
where r* is defined by Eq. 6 as follows. EQU r*=E[f.sub.y (t,.THETA..sub.t.sup.Y)p(X)Z]/(E[f.sub.y (t,.THETA..sub.t.sup.Y)Z]E[p(X)Z]) (6)
where E[] is an expectation value, f.sub.y (t,.THETA..sub.t.sup.Y) is a probability of firing when an internal state and past history of the neuron Y are given.
When estimating a connectivity relation between neurons using the conventional technique of estimation, the participation of the third neuron Z in the connectivity between the two neurons X and Y can be eliminated. In spite of that, the conventional technique has a problem that the technique cannot estimate how the third neuron Z participates in the connectivity between the two neurons X and Y.